Stephen Pizer in “Numerical Computing and Mathematical Analysis” wrote amazingliny easy description of Gauss-Seidel solver.
The most exciting thing for me was pictures which help to understand Guass Seidel iterations on intuitive level.
Pictures show several iterations for 2D system of linear equations.
Two transitions are equal to one iteration of the algorithm.
On the following picture, process converges to exact solution point:
Here one can see how Gauss Seidel fails to find solution:
In general, Gauss-Seidel iterations converge if matrix A is positive definite and symmetric.
very interesting, but what does the two crossing lines represents ? It seems that they are responsible for the convergence.
Two crossing lines represent the system of equations. They are so called hyperplanes.
Here we took 2D system, that means
we have:
a*x + b*y = c
a2*x + b2*y = c2
and we need to obtain such x,y where these two lines intersect each other.
Yes, you are right, these two lines are responsible for the convergence.
This means that Gauss Seidel solver would not converge for some systems and for others it would converge with different speed.
To analyse the ability of convergence for a certain system and its speed one would consider the spectral radius of the iteration matrix for a solving algorithm and the input system.
This is another topic and if you want I would write about it in my next post.
Yes, I would be really interested to learn more about the ability of convergence and the spectral radius.
By the way, your blog is very interesting !
very good